Optimal Distance Labeling Scheme for Interval and Circular-arc Graphs C. Gavoille LaBRI, University Bordeaux I C. Paul CNRS - LIRMM, University of Montpellier II 1 Graph Representations How to encode informations about a graph G in such a way that an oracle is able to answer queries on that graph? Examples of queries : adjacency, connectivity, distance, rout- ing, connectivity . Type of representation: centralized, distributed or mixed Evaluation: a tradeoff between • Space: number of bits needed • Times: complexity to answer queries and to compute the labels (usually the WORD-RAM model is used) 2 Labeling Schemes Let Q be a type of queries (adjacency, distance,. ). For a family F of graphs, a Q-labeling scheme is a pair of func- tions hL, fi such that: • L(v,G) is a binary label associated to vertex v in graph G • f(L(x,G), L(y,G)) is a decoder function that answer the querie bewteen x and y in graph G The labeling scheme is said a l(n)-Q labeling scheme if for every n-vertex graph G ∈ F, the length of the labels is bounded by l(n) bits. 3 Adjacency vs. Distance Labeling Schemes Implicit graph conjecture [Kannan, Naor & Rudich 1992] Any hereditary family F containing no more than 2n.k(n) graphs of n vertices enjoys a O(k(n))-adjacency labeling scheme. 2 log n-adjacency labeling scheme Example of trees 2 ( O(log n)-distance labeling scheme Question: does there exists a non-trivial family of graphs (large enough) for which the distance can be encoded within the same order of space than the adjacency? 4 Interval Graphs A graph is an interval graph if it is the intersection graph of a family of intervals on the real line. g i e i c f j b d f j a d g k b h a c e h k Interval graphs enjoys a 2 log n-adjacency labeling scheme : any vertex x stores its left and right boundaries r(x),l(x) [Katz, Katz & Peleg 2000] The family of interval graphs en- joys an O(log2 n)-distance labeling scheme (DLS for short) 5 Optimal Distance Labeling Schemes An interval graph is proper iff there exists a layout without any interval inclusion. Theorem 1 [Gavoille & Paul 2003] • n vertex proper interval graphs enjoys a 2 dlog ne-DLS; • n vertex interval graphs enjoys a 3 + 5 dlog n)e-DLS; • n vertex circular-arc graphs enjoys a O(log n)-DLS. The distance decoder has constant time decoder and given the sorted list of intervals, the labels can be computed in O(n) time. 6 Layer Partition Let [x0,...xk] be the base-path where x0 has r(x) minimum and ∀i> 0, xi ∈ N(xi−1) such that r(xi) is maximum e i a f i c f j b d g j a d g k b h c e h k V1 V2 V3 V4 The layer partition V1,...Vk is defined by Vi = {v|l(v) < l(xi−1)}\ 06j<i Vj with V0 = ∅ S 7 Graph of Errors Let λ(x) be the integer such that x ∈ Vλ(x) and let H be the digraph on V composed of the arcs xy such that λ(x) < λ(y) and (x,y) ∈ E The graph of errors is the transitive closure Ht of H, let t adjHt(x,y)=1 iff xy is an arc of H 8 Theorem 2 For all distinct vertices x, y such that λ(x) 6 λ(y), distG(x,y)= λ(y) − λ(x) + 1 − adjHt(x,y) Theorem 3 There exists a linear ordering π of the vertices, con- structible in O(n) time, such that adjHt(x,y) = 1 iff λ(x) < λ(y) and π(x) >π(y) π is the pop ordering of a DFS on H using l(x) as a priority rule. The label of the vertex x is L(x,G)= hλ(x),π(x)i 9 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π Stack: a 10 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 Stack: /a 11 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 Stack: /a b 12 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 Stack: /a b d 13 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 Stack: /a b d f 14 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 Stack: /a bdfi 15 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 2 Stack: /a b d f /i 16 An Example e i a fi c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 3 2 Stack: /a b d /f /i 17 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 4 3 2 Stack: /a b /d /f /i 18 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 2 Stack: /a /b /d /f /i 19 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 2 Stack: /a /b /d /f /i c 20 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 2 Stack: /a /b /d /f /i c e 21 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 2 Stack: /a /b /d /f /i c e g 22 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 2 Stack: /a /b /d /f /i cegj 23 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 2 6 Stack: /a /b /d /f /i c e g /j 24 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 7 2 6 Stack: /a /b /d /f /i c e /g /j 25 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 7 2 6 Stack: /a /b /d /f /i c e /g /j h 26 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 7 2 6 Stack: /a /b /d /f /i c e /g /j h k 27 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 3 7 2 6 8 Stack: /a /b /d /f /i c e /g /j h /k 28 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 8 3 7 9 2 6 8 Stack: /a /b /d /f /i c e /g /j /h/k 29 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 4 10 3 7 9 2 6 8 Stack: /a /b /d /f /i c /e /g /j /h/k 30 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 11 4 10 3 7 9 2 6 8 Stack: /a /b /d /f /i /c /e /g /j /h /k 31 An Example e i a f i c f j b d g j a d g k b h c e h k vertex a b c d e f g h i j k λ 1 1 1 2 2 3 3 3 4 4 4 π 1 5 11 4 10 3 7 9 2 6 8 dist(b, i)= λ(i) − λ(b) + 1 − adjHt(b, i) = 4 − 1 + 1 − 1 = 3 dist(d, k)= λ(k) − λ(d) + 1 − adjHt(d, k) = 4 − 2 + 1 − 0 = 3 32 Result for proper interval graphs Theorem 4 The family of n-vertex proper interval graphs enjoys a distance labeling scheme using labels of length 2 dlog ne bits and the distance decoder has a O(1) time complexity.